The biharmonic hypersurface flow and the Willmore flow in higher dimensions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915760887037952 |
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| author | Fu, Yu Hong, Min-Chun Tian, Gang |
| author_facet | Fu, Yu Hong, Min-Chun Tian, Gang |
| contents | The biharmonic flow of hypersurfaces $M^n$ immersed in the Euclidean space $\mathbb {R}^{n+1}$ for $n\geq 2$ is given by a fourth order geometric evolution equation, which is similar to the Willmore flow. We apply the Michael-Simon-Sobolev inequality to establish new Gagliardo-Nirenberg inequalities on hypersurfaces. Based on these Gagliardo-Nirenberg inequalities, we apply local energy estimates to extend the solution by a covering argument and obtain an estimate on the maximal existence time of the biharmonic flow of hypersurfaces in higher dimensions. In particular, we solve a problem in \cite{BWW} on the biharmonic hypersurface flow for $n=4$. Finally, we apply our new approach to prove global existence of the Willmore flow in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The biharmonic hypersurface flow and the Willmore flow in higher dimensions Fu, Yu Hong, Min-Chun Tian, Gang Differential Geometry The biharmonic flow of hypersurfaces $M^n$ immersed in the Euclidean space $\mathbb {R}^{n+1}$ for $n\geq 2$ is given by a fourth order geometric evolution equation, which is similar to the Willmore flow. We apply the Michael-Simon-Sobolev inequality to establish new Gagliardo-Nirenberg inequalities on hypersurfaces. Based on these Gagliardo-Nirenberg inequalities, we apply local energy estimates to extend the solution by a covering argument and obtain an estimate on the maximal existence time of the biharmonic flow of hypersurfaces in higher dimensions. In particular, we solve a problem in \cite{BWW} on the biharmonic hypersurface flow for $n=4$. Finally, we apply our new approach to prove global existence of the Willmore flow in higher dimensions. |
| title | The biharmonic hypersurface flow and the Willmore flow in higher dimensions |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2505.19727 |